Diagram (mathematical logic)

In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.

Let ${\displaystyle {\mathcal {L}}}${\displaystyle {\mathcal {L}}} be a first-order language and ${\displaystyle T}${\displaystyle T} be a theory over ${\displaystyle {\mathcal {L}}.}${\displaystyle {\mathcal {L}}.} For a model ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}} of ${\displaystyle T}${\displaystyle T} one expands ${\displaystyle {\mathcal {L}}}${\displaystyle {\mathcal {L}}} to a new language

${\displaystyle {\mathcal {L}}_{A}:={\mathcal {L}}\cup \{c_{a}:a\in A\}}${\displaystyle {\mathcal {L}}_{A}:={\mathcal {L}}\cup \{c_{a}:a\in A\}}

by adding a new constant symbol ${\displaystyle c_{a}}${\displaystyle c_{a}} for each element ${\displaystyle a}${\displaystyle a} in ${\displaystyle A,}${\displaystyle A,} where ${\displaystyle A}${\displaystyle A} is a subset of the domain of ${\displaystyle {\mathfrak {A}}.}${\displaystyle {\mathfrak {A}}.} Now one may expand ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}} to the model

${\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a)_{a\in A}.}${\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a)_{a\in A}.}

The positive diagram of ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}}, sometimes denoted ${\displaystyle D^{+}({\mathfrak {A}})}${\displaystyle D^{+}({\mathfrak {A}})}, is the set of all those atomic sentences which hold in ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}} while the negative diagram, denoted ${\displaystyle D^{-}({\mathfrak {A}}),}${\displaystyle D^{-}({\mathfrak {A}}),} thereof is the set of all those atomic sentences which do not hold in ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}}.

The diagram ${\displaystyle D({\mathfrak {A}})}${\displaystyle D({\mathfrak {A}})} of ${\displaystyle {\mathfrak {A}}}${\displaystyle {\mathfrak {A}}} is the set of all atomic sentences and negations of atomic sentences of ${\displaystyle {\mathcal {L}}_{A}}${\displaystyle {\mathcal {L}}_{A}} that hold in ${\displaystyle {\mathfrak {A}}_{A}.}${\displaystyle {\mathfrak {A}}_{A}.}^{[1] [2]} Symbolically, ${\displaystyle D({\mathfrak {A}})=D^{+}({\mathfrak {A}})\cup \neg D^{-}({\mathfrak {A}})}${\displaystyle D({\mathfrak {A}})=D^{+}({\mathfrak {A}})\cup \neg D^{-}({\mathfrak {A}})}.

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